1.

Examine the continuity of the following function at given point : f(x)=(log x-log 8)/(x-8)", for "x ne 8 "8,for "x=8 "at,"x=8

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SOLUTION :Given `f(8)=8"…(i)"`
`UNDERSET(xrarr8)(lim)f(x)=underset(xrarr8)(lim)(logx-LOG8)/(x-8)`
Putting `x=8+h,` then `x-8=h` and as `xrarr8, hrarr0.`
`therefore""underset(xrarr8)(lim)f(x)=underset(hrarr0)(lim)(log(8+h)-log8)/(h)`
`=underset(hrarr0)(lim)(log((8+h)/(8)))/(h)`
`=underset(hrarr0)(lim)(log(1+(h)/(8)))/((h)/(8))xx(1)/(8)`
`=(1)/(8)XX1(because underset(xrarr0)(lim)(log(1+x))/(x)=1)"...(ii)"`
From equation (i) and (ii),
`underset(xrarr8)(lim)f(x) ne f(8)`
`therefore"f is discontinuous at x = 8."`


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